Title: The 2-envelope paradox
The 2-envelope paraox is a classical probability conundrum. Two players each receive an envelope: one containing the amount a and the other 2a. By turns, each player may either keep the amount received or switch envelopes. The conundrum is that if a player assumes the envelopes are equally likely, it is always best to switch, which seems paradoxical. However, having observed one value, the problem becomes essentially one of hypothesis testing based on a single observation, and the conditional probabilities will generally fail to be equally likely. Thus, the player must condition on the observed value, making the problem one of standard statistical inference, and not paradoxical. Here we take a general nonparametric approach and consider finding the envelope containing the larger value of the expectation of any specified function, v(x). The basic result is that if v(x) is bounded, then there is a randomized rule (called a "Blackwell pointer" by some colleagues) under which the success probability for choosing the larger envelope is greater than 1/2, uniformly over the set of all distribution-pairs. Other criteria are considered and counter-examples are presented to show that more general conditions will not provide success probabilities greater than 1/2.
